Congruence II Proposition List
This topic proves the well-known Alternate Interior Angle and Exterior Angle Propositions much as Euclid did for the Euclidean plane.
However, Euclid had gaps in these proofs; in fact, his gaps may not be noticed until trying to follow his proofs on a sphere.
This topic uses the Crossbar Proposition to close the gap in the Exterior Angle Proposition.
Also the SAA Congruence Criterion and the Hypotenuse-Leg Congruence Criterion are detailed.
Propositions concerning midpoints, bisectors, and relations between sides and angles of a triangle are proven.
Proposition (Alternate Interior Angle) (i) If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines are parrallel. (ii) Two lines perpendicular to the same line are parallel. (iii) If
is any line and
is any point not on
there exists at least one line
through
parallel to
Proposition (Exterior Angle Property) An exterior angle of a triangle is greater than either remote interior angle.
Proposition (SAA Congruence Criterion) Given
and
Then
Proposition (Hypotenuse-Leg Criterion) Two right triangles are congruent if the hypotneuse and a leg of one are congruent repsectively to the hypotenuse and a leg of the other.
Proposition (Midpoints) Every segment has a unique midpoint.
Proposition (Bisectors) (i) Every angle has a unqiue bisector (ii) Every segment has a unique perpendicular bisector.
Proposition (Larger Angle Larger Side) In a triangle
the greater angle lies opposite the greater side and the greater side lies opposite the greater angle.
Proposition (Side Angle Comparison) Given
and
if
and
then
if and only if
Proposition (Segment Comparison) (i) If
and
then
![congruence ii proposition list _gr_20.gif]](pages/congruence-ii-proposition-list/Images/congruence-ii-proposition-list_gr_20.gif) (ii) Given any triangle
and point
such that
then
or
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Cite this as: Congruence Ii Proposition List Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/congruence-ii-proposition-list.html
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